Contents

Conceptual and Historical Foundations

In the years just prior to Einstein's publication of his special relativity theory Dutch physicist Hendrik Lorentz showed using Maxwell's equations that the electric fields within an object of distributed charge density would produce inertia in proportion to the object's electric potential energy content divided by the square of the speed of light. This production of inertia from field energy results from the curvature of the electric field lines of force between the elements of charge in an accelerated body which the $1/c^2$ corrections to Maxwell equations' electrostatics generate. An important and historic manifestation of this electric energy contribution to mass is one of the terms in the Weizsäcker-Bethe semi-empirical mass formula for nuclei which leads to favorable energetics for nuclear fission of nuclei with sufficiently large atomic number $Z$:
\[
\delta M(A,Z)=\frac{3}{5} \frac{Z(Z-1)e^2}{c^2 R(A)}
\]
for $Z$ uniformly distributed protons within a nucleus of $A$ baryons and radius $R(A)=R_o \: A^{1/3} $, $R_o$ being a fixed length parameter related to nuclear matter's central density.

The subsequent theory of special relativity generalized this connection by asserting that every form of energy within a body contributes to the body's inertial mass; $M=E/c^2$. This leads naturally to the question of whether the gravitational binding energy within celestial bodies contribute to those bodies' masses in accord with the relationship from special relativity? This question has an additional interest for gravity. There are two masses for every body in gravitational dynamics -- the inertial masses of each body and the so-called gravitational masses (gravitational coupling strengths) of the bodies indicating their strength for both producing and responding to gravitational fields or forces. Newton set the gravitational mass of bodies equal to their inertial masses in response to the observations of Galileo, himself, and others that all objects seemed to fall at identical rates in gravity, but for our considerations of the conversion of internal gravitational energy into mass the two mass concepts must be kept apart. Different aspects of the gravitational field theory are involved in determination of the internal gravitational energy contribution to each type of mass. One must also investigate whether and how strongly gravitational fields of outside bodies "pull on" (couple to) the gravitational energy within bodies and the resulting contributions to the gravitational to inertial mass ratio of celestial bodies. This leads to a plausible hypothesis for both theoretical and experimental exploration of whether celestial bodies might have a gravitational to inertial mass ratio which differs from one in proportion to each body's Newtonian gravitational binding energy,
\begin{equation}
\tag{1}
\frac{M(G)_i }{M(I)_i}=1-\eta \frac{1}{M_i c^2} \int \frac{G\rho(\vec{r})\rho (\vec{r}' )\:d^3 rd^3 r'}{2\:|\vec{r}-\vec{r} '|}
\end{equation}
thereby producing the possibility of novel and measurable alterations of body trajectories in celestial mechanics:
\[
\frac{d^2\vec{r}_i}{dt^2}=\frac{M(G)_i}{M(I)_i}\sum_j \frac{GM(G)_j}{\left|\vec{r}_i-\vec{r}_j \right|^3} (\vec{r}_j-\vec{r}_i) + \cdots
\]
Even if the coefficient $\eta$ in Equation (1) has the zero value of general relativity, it could very well have non-zero value in alternative theories of gravity, rendering the measurement of this parameter for celestial bodies a new route for testing gravitational theory. And if the coefficient $\eta$ were empirically found to be zero, the aspects of general relativity's equations of motion for bodies which were ingredients in determining that $\eta = 0$ could very well be broader in scope than those limited features of the theory which were relevant to the traditional tests of general relativity. In the late 1960s the experimental foundations of general relativity consisted of the anomalous perihelion precession of the planet Mercury's orbit, and the deflection of light rays from distant stars which made close passage by the Sun and were viewed during solar eclipses. Additionally, laboratory experiments comparing the free fall rates of different elements had concluded that any material dependence of those rates was less than a part in $10^{11}$. But since those laboratory samples contained utterly negligible fractions of gravitational binding energy (about a part in $10^{25}$), those experiments did not speak to the hypothesis given in Equation (1). They just showed that all forms of field energies other than gravitational -- nuclear, electromagnetic, etc., -- as well as kinetic energies in the various elements contributed equally to high precision in producing gravitational and inertial masses of material matter.

Eddington had shed transparency on two features of the static spherically symmetric metric field of the Sun which were measured by the light deflection and Mercury perihelion precession observations when he expressed the Sun's static metric potentials as (Eddington A, 1957)
\begin{eqnarray}
\tag{2}
g_{00} &=& 1-2\frac {Gm}{c^2 r}+2\beta \left( \frac {Gm}{c^2 r} \right)^2 \\
-g_{ab} &=& \left( 1+2\gamma \frac {Gm}{c^2 r} \right)\delta_{ab} \nonumber
\end{eqnarray}
with $\gamma =\beta=1$ in general relativity but possibly having different values in alternative metric theories of gravity. Then the two mentioned observations could be interpreted as measuring those parameters through the calculated relationships:
\[
\omega_P=(2+2\gamma -\beta )\frac{GM(Sun)}{c^2 R(1-e^2)}\frac{2\pi}{T(Mer)} \qquad
\Theta=2(1+\gamma )\frac{GM(Sun)}{c^2 D(Sun)}
\]
with $R$ and $e$ being the semi-major axis and eccentricity of Mercury's orbit and $T(Mer)$ being the orbit's period. $D(Sun)$ is the distance of closest approach of the deflected light ray passing the Sun. The calculations of these observational effects rested on treating the light ray as a test particle moving on a null geodesic through the Sun's metric field as parameterized by Eddington - Equation (2):
\[
\sqrt{g_{\mu \nu}\:dx^{\mu} dx^{\nu}}=0
\]
which yields a variable coordinate speed of light,
\[
c(r)=c_{\infty}\left(1-(1+\gamma)\frac{GM(Sun)}{c^2 r} \right)
\]
while Mercury's orbit was assumed to be a test particle moving on a geodesic (extremum trajectory) of the same gravitational metric field.
\[
\frac{d}{dt}\frac{\partial L}{\partial \vec{v}}-\frac{\partial L}{\partial \vec{r}}=0 \text{ with } L=\sqrt{g_{\mu \nu}\:dx^{\mu}/dt\: dx^{\nu}/dt}
\]

But the calculation of the gravitational to inertial mass ratio of a celestial body, taking into account its internal gravitational energy, depends on a much more general and complete expression for the gravitational metric fields produced by a general N-body system of moving sources, thereby revealing that broader features of the gravitational metric field contribute to this ratio and its observable effects, as well as involving the Eddington parameters $\gamma$ and $\beta$ in a novel combination. The parameter $\beta$ indeed acquires a more precise experimental measurement from observables dependent on this ratio.

Calculation of a Celestial Body's Gravitational to Inertial Mass Ratio

In order to assume as little as possible about the details of the non-gravitational interactions between the matter elements of a celestial body, the original calculation of gravitational to inertial mass for such a body was done by considering it as an equilibrium gas held together gravitationally; N "atoms" or mass elements were assumed to be held together in stationary equilibrium by a balance between attractive gravitational forces and the effective pressure of the atoms' kinetic motion. Gravity was assumed to be a metric theory in which a tensor field $g_{\mu \nu}$ was the final vehicle expressing gravity's effect on matter. Each atom was assumed to follow the geodesic motion through the metric field as given below by Equation (3). The metric field components are produced by both a distant external body (or bodies) and the other N-1 atoms of the celestial body being considered. A quite general metric field expansion for the metric fields was needed to replace the static, spherically symmetric, single body source, special case of Eddington. The source of the metric field was now a many body system with the individual sources (the atoms or mass elements) being in motion and undergoing acceleration rather than being at rest. The generalized metric field expansion to first post-Newtonian order takes the form
\begin{eqnarray}
\tag{3}
g_{00}&=& 1-2U +2 \beta U^2 +2\alpha ' \sum_{ij} \frac {G^2m_im_j }{c^4 |\vec{r}-\vec{r}_i ||\vec{r}_i -\vec{r}_j |} \\
&-&\alpha ''\sum_i \frac {Gm_i}{c^4 |\vec{r}-\vec{r}_i |}v_i ^2+
\alpha ''' \sum_i \frac {Gm_i}{c^4 |\vec{r}-\vec{r}_i |^3 }(\vec{v}_i \cdot (\vec{r}-\vec{r}_i ))^2 +
\chi \sum_i \frac {Gm_i}{c^4 |\vec{r}-\vec{r}_i |} \vec{a}_i \cdot (\vec{r}-\vec{r}_i ) \nonumber \\
-g_{ab} &=&( 1+2 \gamma U )\delta_{ab}\mbox{ for }a,b=x,y,z \nonumber \\
g_{0a}&=&4 \Delta \sum_i \frac {Gm_i}{c^3 |\vec{r}-\vec{r}_i |}(\vec{v}_i )_a +4\Delta '\sum_i \frac {Gm_i }{c^3 |\vec{r}-\vec{r}_i |^3}
\vec{v}_i \cdot (\vec{r}-\vec{r}_i )(\vec{r}-\vec{r}_i )_a \nonumber
\end{eqnarray}
with an N-body Newtonian potential replacing Eddington's single source $m$:
\[
U=\sum_i \frac {Gm_i}{c^2 |\vec{r}-\vec{r}_i |}
\]
Note the multiplicity of potentials now required to take into general account the motion of the matter sources, and especially the mixed space-time potentials $g_{0a}$.
And a novel nonlinear potential parameterized by $\alpha '$ supplements the $\beta$ potential by necessity in the temporal $g_{00}$ metric potential when multiple sources are present.
A potential proportional to source accelerations $\vec{a}_i$ is included, although coordinate gauge choices could make such a potential absent. At this stage minimum assumptions are employed to restrict the form of the general metric field expansion; neither gauge considerations nor presence or not of energy-momentum conservation laws of isolated systems, nor even the Lorentz invariance of resulting gravitational physics are imposed on the metric field expansions, because we want to consider the possibility that an alternative theory might not fulfill those conditions. The various coefficients $\gamma,\beta,\Delta,\chi, \, \dots, $ are tags used to help keep track of how each metric field potential ultimately contributes to any calculated experimental observable. Those numerical tags are calculable within any metric theory of gravity and generally will differ from the values they take in Einstein's pure tensor general relativity. Given this general metric expansion, each atom $i$ of the celestial body then acquires its equation of motion from the previously stated geodesic principle which takes the explicit form
\begin{equation}
\tag{4}
\left(\frac {d}{dt}\frac {\partial }{\partial \vec{v}_i}-\frac {\partial }{\partial \vec{r}_i}\right) \sqrt{g_{00}+2\vec{h}\cdot \vec{v}_i -\left( 1+2\gamma U \right) v_i ^2}=0
\end{equation}
with $g_{00}$ and $\vec{h}=(g_{0x},g_{0y},g_{0z})$ given in Equations (3). The key to obtaining the modification of a body's total gravitational mass due to its internal gravitational binding energy is the realization that each atom $i$ of the body responds to nonlinear gravitational fields due to and proportional to both the external bodies and the other matter within the body of interest, itself. With
\[
U = \frac{1}{c^2}\left(\frac{GM_{ex}}{|\vec{R}-\vec{r}_i |}+\sum_{j \neq i\:in\:body} \frac{Gm_j}{r_{ij}}\right)
\]
two types of explicitly nonlinear gravitational potentials in the Lagrangian for atom $i$ each produce parts which are additive accelerations for the body as a whole
when collected in a weighted sum over all atoms of the body: these contributing parts have the forms
\begin{eqnarray}
U^2\rightarrow2\frac{Gm_j }{c^2r_{ij}} \frac{GM_{ex }}{c^2|\vec{R}-\vec{r}_i |}\rightarrow 2\frac{Gm_j }{c^4 r_{ij}}
\vec{g}_{ex}\cdot \vec{r}_i \nonumber \\
\sum_{qp}\frac{G^2m_p m_q}{r_{ip} r_{pq}}\rightarrow \frac{Gm_j }{r_{ij}} \frac{GM_{ex }}{|\vec{R}-\vec{r}_j |}\rightarrow
\frac{Gm_j }{r_{ij}}\vec{g}_{ex}\cdot \vec{r}_j \nonumber
\end{eqnarray}
with the gravitational acceleration field of an external body located at $\vec{R}$ being the traditional Newtonian expression
\[
\vec{g}_{ex}=\frac{GM_{ex}}{ R^3}\vec{R}
\]
or the sum of such Newtonian acceleration fields due to multiple bodies.

On organizing the total equation of motion of a body's mass elements, a first category of terms which are collected together are those proportional to both $\vec{g}_{ex} /c^2$ and internal energy-related variables and parameters of the celestial body under consideration. Collectively, such terms contribute to the gravitational mass of the body and are proportional to either kinetic energies of atoms in the body or gravitational potential energies between pairs of atoms in the body. Then setting the acceleration of each atom equal to an internal part, $\vec{a}(int)_i$, plus an acceleration $\vec{A}$ common to all atoms of the body, $\vec{a}_i =\vec{a}(int)_i +\vec{A}$, a second category of terms emerge -- those proportional to $\vec{A} /c^2$ -- which when collected all together contribute to the inertial mass of the body. Just like the collection of terms contributing to gravitational mass, this second collection will be proportional to either kinetic energies of atoms or gravitational potential energies between pairs of atoms within the body. By weighting each atom's equation of motion by the factor $m_i /\sum_j m_j$, and summing over all atoms' equations of motion, all terms other than those collected by the two categories described above cancel out1. The modified Newtonian acceleration of the celestial body then takes the form

\[
\vec{A} = \left [ \frac{M(G)}{M(I)} \right]\vec{g}_{ex}
\]
in which the desired gravitational to inertial mass ratio of the body is given by (Nordtvedt K, 1968)

There is an interesting alternative interpretation of this expression for gravitational to inertial mass ratio which ties this ratio to another novelty of alternative theories of gravity. If $ 4\beta - 3 - \gamma$ is not zero, then Newton's gravitational parameter $G$ will generally vary in space and time depending on the matter distribution which surrounds a local system of interest, such as galaxy or universe surrounding the solar system, or Sun in vicinity of earth-moon system. Although there is no a-priori value established for $G$ in general relativity from which deviations could be measured, and in fact there is only the ability to measure the value of $G$ with modest precision anyway, a spatial variation of $G$ has consequences and is an explanation for the alteration of the gravitational to inertial mass ratios of celestial bodies. Such gradients produce anomalous accelerations on celestial bodies because the gravitational binding energy contributions to their mass-energies proportional to $G$ then vary with position along the gradient of $G$:

\[
\frac{\delta G}{G} = (4\beta - 3 - \gamma ) \frac{GM_S}{c^2R}
\]
The equilibrium gas model for the celestial body used for obtaining this result was subsequently generalized to a solid state model with both electric and gravitational forces between the body's mass elements (Nordtvedt K, 1970) and also to a general fluid model of body matter (Will C, 1971).

Lunar Laser Ranging as measurement of $M(G)/M(I)$

While a star like the Sun has fractional gravitational binding energy of about $4~10^{-6}$ and neutron stars have gravitational binding energies contributing of order ten percent (negatively) to their masses, when taking into account the precision of available means to measure orbits, the lunar orbit has turned out to be the primary tool for measuring the gravitational to inertial mass ratio of celestial bodies. This capability became possible by the placement of a passive laser reflector on the lunar surface during the first Apollo landing in 1969 and subsequent reflector placements with later manned and unmanned lunar landings. Within weeks of the first deployment of a reflector on the Moon, round trip transit times for laser pulses initiated from and returned to observatories on Earth began to be recorded. Laser ranging to reflectors on the Moon has continued using constantly improved technology ever since; initial range precisions of tens of centimeters have been reduced to centimeter-level, and with formal errors of model parameter fits down to the millimeter level precision. But with consideration of overall modeling uncertainties measurement of orbital perturbation amplitudes at the several millimeter level are today obtained. The key frequencies of lunar orbital motion are determined with even higher precision due to the almost half century series of range measurements spanning hundreds of lunar orbit cycles. Almost 20,000 range measurements between stations on Earth and reflectors on the Moon have been made between 1969 and the present (Shapiro I I, Counselman, King R W, 1976; Williams J G et al, 1976; Williams J G, Turyshev S G, Boggs D H, 2012).

If the difference between acceleration of Moon and Earth in the Sun's gravitational field $\vec{g}_S$ is
\[
\vec{a}_M - \vec{a}_E = \left( \Delta_M - \Delta_S \right) \vec{g}_S = \Delta \vec{g}_S
\]
then the lunar orbit will be "polarized" toward or away from the Sun, depending on sign of $\Delta$. In the simplest approximation the radial Moon-Earth orbital distance $r$ and orbit's angular motion $h=|\vec{r} \times \vec{v} |$ fulfill the perturbed equations of motion:
\begin{eqnarray}
\ddot{r} = -\frac{G m_E}{r^2} + \frac{h^2}{r^3} = \Delta g_S \cos \dot{D}t \nonumber \\
\dot{h} = -r \Delta g_S \sin \dot{D}t \nonumber
\end{eqnarray}
with $\dot{D}$ being the Moon's synodic frequency (frequency of New Moon occurrences). In this approximation a synodic frequency perturbation in Moon-Earth range then results,
\[
\delta r(t) =\Delta \left(1 + \frac{2\omega}{\dot{D}}\right) \frac{g_S}{\omega^2 - \dot{D}^2 } \cos \dot{D}t \approx 1.8 \:10^{12} \Delta \cos \dot{D}t\text{ cm}
\]
with $\omega$ being the Moon's orbital frequency (Nordtvedt K 1968). The closeness of the lunar synodic frequency $\dot{D}$ to the orbital frequency $\omega$ gives a strong resonance enhancement to the polarization. A more careful calculation of the sensitivity of the lunar orbit to such a perturbation takes into account the Sun's Newtonian tidal distortion of the orbit. This tidal distortion further enhances the resonance of the synodic perturbation by lowering the Moon's anomalistic resonance frequency $\dot{A}$, but more importantly the tidal perturbation of frequency $2\dot{D}$ which Newton called the $lunar\; variation$ produces strong positive feedback of the synodic perturbation ($\cos\dot{D}t \cos 2\dot{D}t \rightarrow \cos \dot{D}t + \cos 3\dot{D}t $) which almost doubles the orbit's sensitivity to a non-zero $\Delta $ (Nordtvedt K, 1994). The $1.8\:10^{12} \Delta \cos \dot{D}t \text{ cm}$ sensitivity consequently becomes about $3.3\:10^{12} \Delta \cos \dot{D}t \text{ cm}$. Since the fractional gravitational binding energy of Earth is about $4\:10^{-10}$, the $\cos \dot{D} t$ range perturbation amplitude then becomes $1.3\:10^3 \eta \text{ cm}$. The Sun's Newtonian octupolar tidal acceleration of the Moon relative to Earth, proportional to $GM(Sun)r^2 /R^4$ is the main perturbation producing a synodic $\cos \dot{D} t $ variation in the Earth to Moon distance, having amplitude of about $110 \text{ km}$. But all the system's parameters needed to calculate this perturbation are sufficiently well measured from the ranging data to reduce the uncertainty in this Newtonian contribution to sub-millimeter level. However, other intrinsic model limitations related to the Earth and Moon's surfaces and interiors and how they respond to the various tidal perturbations produce uncertainties in the synodic amplitude at the few millimeter level.

With $\eta = 4\beta - 3 - \gamma$, and $\gamma$ presently constrained to its general relativity value of one within $\pm 2.3\:10^{-5}$ by measurements of the Sun's relativistic time delay of signals from the Cassini spacecraft when its line of sight passed close by the Sun (Bertotti B, Iess L, Tortora P, 2003), the fit of the lunar orbit produces about a part in $10^4$ constraint on the nonlinearity $\beta$ coefficient of Eddington, presently being the best measure of general relativity's nonlinear structure. An alternative way to summarize this result is that the Earth's gravitational binding energy contributes equally to both its gravitational mass and inertial mass at better than a part in a thousand precision in accord with pure tensor general relativity.

If Jupiter anomalously accelerates the Sun due to it having an $M(G)/M(I)$ ratio different than one, then the inner planets' orbits will be polarized in the direction of Jupiter which circles the Sun with about an eleven year period. Interplanetary ranging between Earth and Mars seems the most promising way to carry out such an experiment to measure the Sun's ratio (Anderson J D, Gross M, Nordtvedt K L, Turyshev S G, 1996). And recently a neutron star pulsar along with two white dwarf stars were found in a closely bound three body system PSR J0337+1715 (Ransom S M et al, 2014). With the neutron star of that dynamical system having fractional gravitational binding energy of order $0.1$ it may be possible to use the pulsar arrival times to measure the neutron star's gravitational to inertial mass ratio at a level which probes gravity's nonlinearity to a deeper level than that so far achieved with LLR.

Footnotes

1 An exception are terms proportional to $M_{ex} ^2 /c^2$ which simply represents the nonlinear $1/R^3$ gravitational acceleration that any particle or body experiences towards external bodies. This acceleration already seen and measured from the perihelion precession of Mercury's orbit is extraneous to the question of gravitational to inertial mass ratio of bodies in response to the Newtonian acceleration of the external world.

2 The derivation for this ratio for celestial bodies and suggestion to measure it by lunar laser ranging was dubbed "The Nordtvedt Effect" by colleagues of the author. Due to higher order nonlinearities of gravity theories which deviate from general relativity, gravitational mass of a celestial body could have additional higher order contributions such as

This change in strength of the $1/R$ potential between Sun and Earth is of order a couple parts in $10^{15}$, and presently beyond observability.

3 Viral contributions to inertia are not unique to gravity; in electrodynamics the inertial mass of a system of atoms held together by electric forces and their motions includes the spatial tensor virial.

The virial contribution to inertia can be turned on so to speak by application of external forces on a system; this results, for example, in the inertia of a fluid element becoming $\rho + p/c^2$ with $\rho$ being the actual mass-energy density and $p$ being the pressure on the fluid element. If a celestial body oscillates about its equilibrium the virials will also oscillate about zero; however, for bodies such as Earth such oscillations, driven by tides or earthquakes or atmospheric disturbances are much too small in energy to be detectable. The energy of Earth's rotation is, however, significant, being $ 3\:10^{-13}$ of its mass-energy. In gravity theories which do not fulfill local Lorentz invariance or violate energy-momentum conservation, the gravitational to inertial mass ratio will include spatial tensor contributions proportional to a body's rotational energy (Nordtvedt K, 1969)

4 Using the temporal metric field expression as given by Equation (3) and the conservative condition $\alpha ' = 2 \beta -1$, and letting the Newtonian potential function be a sum of contributions of nearby bodies plus distant surrounding bodies' potential $U_S$,